Use the Division Algorithm to establish the following:
(a) The square of any integer is either of the form 3k or 3k + 1.
(b) The cube of any integer has one of the forms: 9k, 9k + 1, or 9k + 8.
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Step-by-step explanation:
any positive integer is in the. form of 3q,3q+1,3q+2
if n=3q
n^2=(3q)^2
=9q^2
3(3q^2)
is in the form of 3m
m=3q^2
n=3q+1
n^2=(3q+1)^2
=3q^2+1^2+2(3q)(1)
9q^2+1+6q
9q^2+6q+1
3(3q^2+2q)+1
is in the form of 3m+1
m=3q^2+2q
n=3q+2
n^3=(3q+2)^2
3q^2+2^2+2(3q)(2)
9q^2+4+12q
3(3q^2+4q)+4
therefore it is in the form of 3q+4
therefore all positive in refers are in the form of 3q,3q+1 but not in any other form
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